The Zappa-Szép product of twisted groupoids
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913373751345152 |
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| author | Duwenig, Anna Li, Boyu |
| author_facet | Duwenig, Anna Li, Boyu |
| contents | We define and study the external and the internal Zappa-Szép product of twists over groupoids. We determine when a pair $(Σ_{1},Σ_{2})$ of twists over a matched pair $(\mathcal{G}_{1},\mathcal{G}_{2})$ of groupoids gives rise to a Zappa-Szép twist $Σ$ over the Zappa-Szép product $\mathcal{G}_{1}\bowtie\mathcal{G}_{2}$. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ is a C*-blend of its subalgebras corresponding to the subtwists $Σ_{i}\to \mathcal{G}_{i}$. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ of two twists $Σ_{1}\to \mathcal{G}_{1}$ and $Σ_{2}\to \mathcal{G}_{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Zappa-Szép product of twisted groupoids Duwenig, Anna Li, Boyu Operator Algebras 46L55, 46L05, 22A22 We define and study the external and the internal Zappa-Szép product of twists over groupoids. We determine when a pair $(Σ_{1},Σ_{2})$ of twists over a matched pair $(\mathcal{G}_{1},\mathcal{G}_{2})$ of groupoids gives rise to a Zappa-Szép twist $Σ$ over the Zappa-Szép product $\mathcal{G}_{1}\bowtie\mathcal{G}_{2}$. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ is a C*-blend of its subalgebras corresponding to the subtwists $Σ_{i}\to \mathcal{G}_{i}$. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ of two twists $Σ_{1}\to \mathcal{G}_{1}$ and $Σ_{2}\to \mathcal{G}_{2}$. |
| title | The Zappa-Szép product of twisted groupoids |
| topic | Operator Algebras 46L55, 46L05, 22A22 |
| url | https://arxiv.org/abs/2406.00466 |